Word maps and spectra of random graph lifts

We study here the spectra of random lifts of graphs. Let G be a finite connected graph, and let the infinite tree T be its universal cover space. If λ1 and ρ are the spectral radii of G and T respectively, then, as shown by Friedman [Fri03], in almost every n-lift H of G, all “new” eigenvalues of H are ≤ O ( λ 1/2 1 ρ 1/2 ) . Here we improve this bound to O ( λ 1/3 1 ρ 2/3 ) . It is conjectured in [Fri03] that the statement holds with the bound ρ + o(1) which, if true, is tight by [Gre95]. For G a bouquet with d/2 loops, our arguments yield a simple proof that almost every d-regular graph has second eigenvalue O(d). For the bouquet, Friedman [Fri] has famously proved the (nearly?) optimal bound of 2 √ d− 1 + o(1). Central to our work is a new analysis of formal words. Let w be a formal word in letters g 1 , . . . , g ±1 k . The word map associated with w maps the permutations σ1, . . . , σk ∈ Sn to the permutation obtained by replacing for each i, every occurrence of gi in w by σi. We investigate the random variable X (n) w that counts the fixed points in this permutation when the σi are selected uniformly at random. The analysis of the expectation E(X (n) w ) suggests a categorization of formal words which considerably extends the dichotomy of primitive vs. imprimitive words. A major ingredient of a our work is a second categorization of formal words with the same property. We establish some results and make a few conjectures about the relation between the two categorizations. These conjectures suggest a possible approach to (a slightly weaker version of) Friedman’s conjecture. As an aside, we obtain a new conceptual and relatively simple proof of a theorem of A. Nica [Nica94], which determines, for every fixed w, the limit